Solving Inequalities with Exponential Functions

[verty]

I'm wondering how to solve an inequality like this:

x + 3^x < 4

I can see that it is the sum of the identity function and an exponential function. I can see that it is monotone increasing because each of those is. I therefore know that it crosses y=4 at only one x-value (call it x_1) and I know the solution will be x < x_1.

So I need to solve for x in x + 3^x = 4, how would I do that? I see I could draw the graph and read off the value, which I can see is x_1 = 1, but is there a way to calculate it?

Thank you for any clarification.

 

[CRGreathouse]

x + 3^x < 4

Try this: $x+3^x=4+k$ for $k>0$.

 

[acm]

Draw a graph first, let the function be y = x + 3^x - 4 and find where this graph cuts the x-axis (That will satisfy the Ineq, Find an approximate area where the graph cuts the axis, so accuracy is paramount.). From here you should have an idea of the kind of answer you'll need. To find a approximate solution, I'd use Newton's method which is:
x_n+1 = x_n - F(x_n)/F'(x_n)
You can take the first value of x_n from the graph you've drawn.

 

[verty]

Thank you for suggesting Newton's method.

 

[StatusX]

In general, you can't solve equations like a^x+x=b in a closed form, although you can approximate the solution to any degree of accuracy. However in this case, it's easy to check that the solution is exactly x=1.